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A formal Riemannian structure on conformal classes and uniqueness for the $\sigma_2$–Yamabe problem

Matthew Gursky and Jeffrey Streets

Geometry & Topology 22 (2018) 3501–3573

We define a new formal Riemannian metric on a conformal classes of four-manifolds in the context of the σ2–Yamabe problem. Exploiting this new variational structure we show that solutions are unique unless the manifold is conformally equivalent to the round sphere.

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fully nonlinear Yamabe problem, uniqueness
Mathematical Subject Classification 2010
Primary: 58J05
Secondary: 53C44, 58B20
Received: 12 June 2017
Revised: 29 September 2017
Accepted: 10 November 2017
Published: 23 September 2018
Proposed: Simon Donaldson
Seconded: Bruce Kleiner, Tobias H Colding
Matthew Gursky
Department of Mathematics
University of Notre Dame
Notre Dame, IN
United States
Jeffrey Streets
Department of Mathematics
University of California, Irvine
Irvine, CA
United States